>But is it not the case that the original Shor algorithm was specifically about using periodicity for integer factorisation
Periodicity is discrete log.
Discrete log is defined as given a mathematical group G, a generator g in G, and an element h in G, find the power of g that gives h. This is a logarithm, over a discrete set, a finite group. This is equivalent to finding the power of h so h^r = 1, since given either, the extended euclidean algorithm gives an efficient way to get the other number. This number r is called the order of h, and there is no (in general) known classical efficient algorithm to do it.
This can be easy or hard, depending on the group.
Shor period finding is taking a the integers mod N (which is a group, called an abelian group, about the simplest class of groups), and an integer a, and finding the power r of a that is so a^r is 1 mod N. This is exactly the discrete log problem. You found the order of the element h.
Shor is a special, easy case of the more general problem - he solves it efficiently for Abelian cyclic groups, but it has been extended to many, many other groups.
In fact, researchers realizing this was exactly the general problem Shor solved is why all discrete log problems are vulnerable to QC. His method of using a quantum Fourier transform (which has a nice generalization to groups) to find period (which FFT does nicely) is directly applicable to any group. Shor is not some odd outlier algorithm; it is exactly the core (and almost the only) algorithm QC is exponentially faster at.
>But the overwhelming majority of PK crypto in use today (TLS, PGP, the vast majority of E2EE systems) is not secure against quantum attackers. Can we agree on that at least?
Yes, but that is a far cry from your original claim. We only use those systems because they are not widely broken, and like all crypto, if one gets broken, we simply replace. This is not new.
Each is easily replaced by algorithms that quantum cannot touch (unless classical can too), so much so, that many places already use QC resistant algorithms to prevent nation-states messing with them.